CONSTRUCTIVE INTERFERENCE EXPLOITATION FOR DOWNLINK BEAMFORMING BASED ON NOISE ROBUSTNESS AND OUTAGE PROBABILITY. Ka Lung Law and Christos Masouros

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1 CONSTRUCTIVE INTERFERENCE EXPLOITATION FOR DOWNLINK BEAMFORMING BASED ON NOISE ROBUSTNESS AND OUTAGE PROBABILITY Ka Lung Law and Christos Masouros Department of Electrical and Electronic Engineering, University College London, London, UK ABSTRACT Quality of service (QoS is commonly measured in terms of signal to interference plus noise ratio (SINR, where multiuser interference is mitigated in order to improve the performance. As opposed to conventional suppression, interference can be exploited constructively to enhance the desired signal. With the aid of channel state information (CSI at the transmitter and data information, we study symbol-level downlink beamforming problems based on noise robustness and outage probability, respectively, subject to power constraints. We further show that an equivalence relationship between the noise robustness and outage probability symbol-level downlink beamforming problems can be obtained. Finally, we provide an analytic symbol error rate (SER upper bound of the worst user by solving the outage probability-based problem. Our simulations demonstrate that the proposed techniques provide substantial performance improvements over conventional downlink beamforming techniques. Index Terms Downlink beamforming, robust design, error probability, convex optimization, constructive interference. 1. INTRODUCTION In wireless networks, downlink beamforming is an attractive approach as an effective way of simultaneously transmitting an individual data for each user to achieve demand in high data rate. [1, 2]. In addition to the urge for high throughputs and limited power expenses, quality of service (QoS is also a main criterion in modern communications systems. With the knowledge of channel state information (CSI at the transmitter, designing downlink beamformers to improve the QoS for downlink scenario has been studied extensively [3 9]. Zero-forcing (ZF precoding is commonly employed to downlink problem. The multiuser interference signal is nulled in wireless communications [10, 11]. The advantage of ZF precoding is that the algorithm is simple to apply. However, it is not fully optimized. To obtain the optimal solutions, the optimization-based downlink beamforming problems were developed [4, 5, 12 15]. One form of downlink problems is to imize the minimum SINR subject to a total power constraint [4]. The problem is efficiently solved using an iterative algorithm. Taking the CSI mismatch into account, channel robust worst-case downlink beamforming optimization was considered [5, 12 14]. To provide more flexibility than the worst-case scenario, channel outage probability-based downlink beamforming optimization has been introduced [14, 15]. It has been proved that both the worst channel robustness and outage probability-based problems are equivalent. In the SINR-based downlink problem, beamformers are designed to guarantee that the SINR constraints are satisfied. However, the drawback of SINR criteria is that power is wasted by suppressing the interference. Rather than mitigating, one can exploit constructive interference to enhance the useful signal by making use of both the CSI and data information. By exploiting the constructive part of interference to achieve higher performance, the closed-form linear and non-linear precoders were discussed [16 22]. Nonetheless, these precoders are not the optimal design. Optimization-based downlink beamforming precoders by exploiting constructive interference was considered [23, 24]. In line with the above, this paper is based on the symbol-level downlink beamforming optimization by exploiting constructive interference to amplify the signal [23, 24]. In the following analysis, phase-shift keying (PSK modulation is selected. We assume that a time division duplexing (TDD transmission, e.g., downlink channels can be determined by using the knowledge of uplink CSI and uplink-downlink channel reciprocity [25], the availability of perfect CSI at the transmitter and instantaneous data information, as in [23, 24]. We propose a symbol-level downlink beamforming problem based on noise robust design in Section 4 by introducing a geometrical analysis to the optimization problem studied in [23]. We reformulate the optimization to address the symbol-level downlink beamforming problem based on outage probability design in Section 5 by use of duality with the noise robust case. All proposed approaches can be formulated into convex optimizations and can be solved efficiently. We provide an analytic symbol error rate (SER upper bound of the worst user by solving the error probability-based optimization. Notation: E(, Pr(,,, ( ( T, denote statistical expectation, the probability, the absolute value, the Euclidean norm, the complex conjugate, the transpose, respectively. ( and ( are the real part, and the imaginary part, respectively. 2. SYSTEM MODEL AND CONVENTIONAL DOWNLINK BEAMFORMING Let us consider a downlink scenario with a single N-antenna at the base station (BS. We assume that there are K single-antenna users. Letb i be the transmitted data with the unit amplitude of them-order PSK modulation and the given imum angular shift θ = π/m. The transmitted signal at the BS is the N 1vector x = K t ib i, (1 i=1 where t i is the N 1 beamforming vector for the ith user. The received signal for the ith user is given by y i = h T i x+n i, (2 where n i is a complex white Gaussian noise and h i is the N 1 channel vector for the ith user. We present a common downlink /16/$ IEEE 3291 ICASSP 2016

2 θ y i ψ i b i b i θ 1 y i b i (y i b i ψ i (y i b i θ φ i θ, where φ i is an angle such that φ i(x,τ = { tan 1 ( (b i h T i x (b i ht i x τσ (b ih T i x> τσ, 0 b ih T i x = τσ. The disadvantage of (4 are that it is hard to quantify the QoS in terms of τ. In particular, [23] did not provide the relationship between τ and the worst user s SER performance. We address this issue in Section 5. In the next section we present a noise robustnessbased optimization by exploiting the constructive interference. 4. NOISE ROBUST BEAMFORMING OPTIMIZATION (5 a Fig. 1: In M-PSK, (a constructive interference y i within correct detection region; (b vector decomposition of y ib i after rotation by b i. beamforming optimization problem in the literature [4 6], which imizes the minimum SINR subject to a total transmitted power constraint. The problem can be formulated as [4] γ t i,γ s.t. Kj=1 j i h T i t i 2 γ, i=1,...,k, h T j tj 2 +σ2 K t i 2 P 0, (3 i=1 where γ is the minimum SINR and P 0 is the given total transmitted power threshold andσ 2 is the noise variance. 3. CONSTRUCTIVE INTERFERENCE OPTIMIZATION-BASED PRECODING By jointly exploiting the knowledge of the CSI and user data information at the transmitter, the constructive interference-based optimization precoder in [23] improves upon the above conventional optimization. The precoder imizes the shifted distance τ σ of correct detection region away from origin along with the direction of the corresponding transmitted symbol b i by designing the beamformers. The optimal beamformers can guarantee that the resultant received symbolh T i x still falls within the corresponding region. Under the design criterion, the resultant received symbol moves away from the original decision thresholds of the constellation. This leads to an improvement of QoS. The reader interested in additional details of the underlying concept is referred to [23]. The optimization problem can be written in mathematical form as [23] x,τ τ s.t. (b ih T i x ((b ih T i x τσtanθ, x 2 P 0, i=1,...,k, (4 where P 0 is the predefined total transmitted power threshold. The constraints of (4 stem from the fact that the resultant received symbol for the ith user lays on correct detection region, if and only if b In this section, we introduce a noise robust adaptation together with exploiting the constructive interference. First of all, we present an improved systematic treatment of constructive interference for the received signal. For PSK modulation, interference is constructive 1 if the received signal y i lays on the correct detection region, which is the shaded area shown in Fig. 1(a. Under the definition of constructive interference, we obtain the following lemma. Lemma 1. The received signal y i is said to receive constructive interference, if and only if θ ψ i θ (6 whereψ i in Fig. 1(a is the angle between the received signaly i and the transmitted symbol b i such that { ( tan 1 (y i b i ψ i(x,n i (y = i b (y ib i > 0, i (7 0 y ib i = 0. The criterion in (6 can be directly reformulated as the following constraints (y ib i (y ib itanθ 0. (8 Proof. Suppose that the received signal y i is within the correct detection region. To obtain the angle ψ i, we first rotate Fig. 1(a to Fig. 1(b by shifting the constellation by a phase equal to b i, i.e., by multiplying b i. As b i is a unit power, y ib i does not change the magnitude. Then we obtain the inequities in (7 where(y ib i and (y ib i are the projection ofy ib i onto the real and imaginary axis, respectively Noise Uncertainty Radius Maximization The idea of the symbol-level downlink beamforming problem based on noise robustness is to design the beamformers such that the received signal is constructive interference if the noise is within the noise uncertainty set. To improve the noise robustness of the design given the noise variance σ 2, we imize the radiusγσ of the noise uncertainty set such that it can still satisfy the constraints (8 under the power constraint. The noise robustness-based optimization problem by exploiting constructive interference can be written as x,γ Γ s.t. ψ i(x,n i θ, i=1,...,k, n i Γσ x 2 P, (9 1 Note that we consider the resultant received symbol plus noise in our case, while [23] discussed the resultant received symbol h T i x in the formulation. 3292

3 where P is the given total transmit power. By Lemma 1, we rewrite (9 as x,γ Γ s.t. (y ib i (y ib itanθ 0, n i Γσ x 2 P, i=1,...,k. (10 To simplify above problem, we can first solve the inner imization in (10. Corollary 1. For a fixed x, the inner imization in (10 has the following optimal solution as (b ih T i x +Γσ/cosθ (b ih T i xtanθ. (11 Proof. Let ỹ i = b ih T i x+n i. The dual Lagrange function is given by L(κ i,n i = (ỹ i +(ỹ itanθ+κ i( n i 2 Γ 2 σ 2, (12 where κ i 0. Note that (b in i = n Iib Ri n Rib Ii, (13 (b in i = n Iib Ii +n Rib Ri. (14 wheren i n Ri+in Ii, andb i b Ri+ib Ii. Setting L n Ii = 0, we obtain L n Ri = 0 and b Ritanθ +b Iiα i +2κ in Ri = 0, (15a b Riα i +b Iitanθ +2κ in Ii = 0, (15b whereα i = (ỹ i/ (ỹ i anda is the optimal value ofa. If we suppose that κ i = 0, then (15 implies that b Ri = b Ii = 0, which leads to the contradiction. Therefore, we conclude that κ i > 0 and n i 2 = Γ 2 σ 2, (16 by the complementary slackness. Putting (15 into (16 and noticing the fact that b i is a unit power symbol, we obtain κ i = (2Γσcosθ 1. (17 We substitute (17 back into (15, then we get n Ri = (b Ritanθ +b Iiα iγσcosθ, (18a n Ii = (b Riα i b IitanθΓσcosθ. (18b Taking (18 into problem (10, we rewrite the inner imization in (10 as α i(b ih T i x n Ri(b Ritanθ +b Iiα i +n Ii(b Riα i b Iitanθ (b ih T i xtanθ = (b ih T i x +Γσ/cosθ (b ih T i xtanθ, (19 where (ỹ i and (b ih T i x have the same sign because we can assume that the received noise cannot dominate the received signal. According to Corollary 1, we reformulate (10 as a function Γ ( for any givenp 0 such that Γ (P : x,γ Γ s.t. (b ih T i x +Γσ/cosθ (b ih T i xtanθ, x 2 P, i=1,...,k. (20 Problem (20 can be solved using available convex optimization tools [26]. Finally, we obtain the optimal beamformer t i in (1 as t i = x b i/k, (21 where x is the optimal solution in (20. mark 1: Suppose x SD and x NR are optimal solutions of (4 and (20, respectively. Thensinθx NR = x SD. Hence we can treat them as equivalence problems. 5. OUTAGE PROBABILITY APPROACH We assume a noise at the receiver is complex Gaussian with zero mean. In this section, we present a new approach to constructive interference-based downlink beamforming by the noise outage probability. In the concept of noise outage probability, we replace the noise robust downlink beamforming constraints by more flexible probabilistic constraints. We define the noise outage probability for the ith constraint as the probability that received signal lays outside the correct detection region bounded by either the angle θ or θ. The problem can be written as ( min p s.t. Pr π ψ i(x,n i θ p, i=1,...,k, (22a x,p ( Pr π ψ i(x,n i θ p, i=1,...,k, (22b x 2 P. mark 2: Problem (22 and the channel outage probability based downlink beamforming problem in [14, 15] are different. The constraints in [14,15] are outage probabilistic SINR-based with channel random variables, while the constraints are outage probabilistic constructive interference-based with noise random variables. The SER upper bound of the worst user is equal to 2p, which is originated from that the worst case possibility of the received signal laying outside the correct detection region bounded by the angle ±θ is p respectively. It will be shown in the simulation result that the worst user s SER performance calculations close to the upper bound. According to Lemma 1, problem (22 can be expressed as ( min p s.t. Pr (y x,p ib i (y ib itanθ p, (23a ( Pr (y ib i (y ib itanθ p, (23b x 2 P, i=1,...,k. The constraints in (23a and (23b can be rewritten as where Pr(z i +ñ i 0 p, (24 z i = ±(b ih T i x (b ih T i xtanθ, (25 ñ i = ±(b in i (b in itanθ. (26 Asn i is complex Gaussian, we obtain E{(b in i 2 }=E{(b in i 2 }=b 2 σ 2 σ 2 Ri 2 +b2 Ii 2 =σ2 2, (27 E{(b in i(b in i}=b Rib Ii b Rib Ii=0. (28 The variance ofn i is given by E{ñ 2 i} = (1+tan 2 θσ 2 /2 = σ 2 /(2cos 2 θ. (

4 Worst user's SER Conventional [4], K=12 Noise robust ( [23], K=12 Upper bound of noise robust, K=12 Conventional [4], K=10 Noise robust ( [23], K=10 Upper bound of noise robust, K=10 Conventional [4], K=8 Noise robust ( [23], K=8 Upper bound of noise robust, K= Total transmitted power (db Fig. 2: The worst user s SER performance versus transmit power withn = 10. σ Therefore, ñ i N(0, 2cosθ. By ensuring reliable communication link, the noise outage probability must be close to 0. According to [15], we assume that p 0.5. The outage probability constraints in (24 can be expressed in terms of the Gaussian error functionerf( as or equivalently, (b ih T i x + erf 1 (1 2pσ cosθ ( zicosθ 2 erf p, (30 σ (b ih T i xtanθ, i. (31 Hence, the outage probability problem (23 can be written as a functionp ( for any given P 0 such that p (P : min x,p p s.t. (b ih T i x + erf 1 (1 2pσ (b ih T i xtanθ, cosθ x 2 P, i=1,...,k. (32 and the optimal values of (20 and (32 have the following relations: Γ (P = erf 1 (1 2p (P, (33 p (P = erf(γ (P, (34 x p(p = x Γ(P, (35 where x p( P is an optimal solution of (32 for a given power P. 6. SIMULATIONS In our simulations, the system with 4-PSK modulation is considered, i.e., θ = π/4, while it is intuitive that the benefits of the proposed approaches extend to other modulation schemes. The white complex zero-mean Gaussian noise n i is with the variance σ 2 = 1. We Conventional [4], P=5dB Noise robust ( [23], P=5dB Conventional [4], P=15dB Noise robust ( [23], P=15dB Fig. 3: Distribution of received signals on complex plane with N = 10, and K = 10. consider a constructive interference-based downlink beamforming network with N = 10 antennas, while it is obvious that the benefits shown extend to different numbers of antennas. Let ω i be a uniformly distributed random number between π/2 and π/2. We model the downlink channel between the BS and ith user as [27] h i= [ 1,e jπ sinω i,...,e jπ(n 1sinω i] T. (36 We compare two different techniques: Conventional [4] refers to the SINR balancing problem in [4]; Noise robust ( = [23] stands for the problem (20. Note that (20 is equivalent to (4, which is proposed in [23]. Upper bound of noise robust stands for the SER upper bound of the worst user by solving noise robust approach and it is equal to 2p according to mark 2, where p is the outage probability of (32. Since we have shown in Section 5 that the noise robust approach of (20 and the outage probability approach of (32 are equivalent, we only consider the noise robust approach in the following simulations. Fig. 2 compares the worst user s SER performance for the different techniques. In Fig. 2, we fix the number of users and compare the worst user s SER performance of our proposed approaches and the conventional approach of [4] versus the total transmitted power P with different numbers of user K. It can be seen from the figure that the noise robust approach outperforms the conventional method of (3. Furthermore, the worst user s SER performance calculations of the proposed noise robust approach match close to the SER upper bound. Fig. 3 displays the distribution of the received signals using the two techniques on complex plane with P = 5dB and P = 15dB. Here, we set the transmitted symbol to be 1. The right side of dotted line is the constructive area of the constellation. Therefore, the received signals are valid if they lay on the right side behind the dotted line. We observe from Fig. 3 that the received signals of our proposed method can better lay on the correct detection region compared to the conventional method. Moreover, We notice that when the power increases, our technique can shift the received signals further away from the decision threshold than the conventional technique. 3294

5 7. REFERENCES [1] E. Dahlman, S. Parkvall, and J. Skold, 4G: LTE/LTE-Advanced for Mobile Broadband: LTE/LTE-Advanced for Mobile Broadband, Elsevier Science, [2] A. F. Molisch, Wireless Communications, John Wiley and Sons Ltd, [3] F. Rashid-Farrokhi, K.J.R. Liu, and L. Tassiulas, Transmit beamforming and power control for cellular wireless systems, Selected Areas in Communications, IEEE Journal on, vol. 16, no. 8, pp , Oct [4] M. Schubert and H. Boche, Solution of the multiuser downlink beamforming problem with individual SINR constraints, IEEE Transactions on Vehicular Technology, vol. 53, no. 1, pp , Jan [5] M. Bengtsson and B. Ottersten, Optimal downlink beamforming using semidefinite optimization, in Annual Allerton Conference on Communication, Control and Computing, 1999, vol. 37, pp [6] Mats Bengtsson and Björn Ottersten, Optimal and suboptimal transmit beamforming, Handbook of Antennas in Wireless Communications, [7] A Wiesel, Y. C. Eldar, and S. Shamai, Linear precoding via conic optimization for fixed MIMO receivers, IEEE Transactions on Signal Processing, vol. 54, no. 1, pp , Jan [8] A. B. Gershman, N. D. Sidiropoulos, S. Shahbazpanahi, M. Bengtsson, and B. Ottersten, Convex optimization-based beamforming: From receive to transmit and network designs, IEEE Signal Processing Magazine, vol. 27, no. 3, pp , May [9] Jinho Choi, Downlink multiuser beamforming with compensation of channel reciprocity from RF impairments, Communications, IEEE Transactions on, vol. 63, no. 6, pp , June [10] Q.H. Spencer, A.L. Swindlehurst, and M. Haardt, Zeroforcing methods for downlink spatial multiplexing in multiuser MIMO channels, Signal Processing, IEEE Transactions on, vol. 52, no. 2, pp , Feb [11] Kai-Kit Wong, R.D. Murch, and K.B. Letaief, A joint-channel diagonalization for multiuser mimo antenna systems, Wireless Communications, IEEE Transactions on, vol. 2, no. 4, pp , [12] M.B. Shenouda and T.N. Davidson, Convex conic formulations of robust downlink precoder designs with quality of service constraints, Selected Topics in Signal Processing, IEEE Journal of, vol. 1, no. 4, pp , Dec [13] N. Vucic and H. Boche, Robust QoS-constrained optimization of downlink multiuser MISO systems, Signal Processing, IEEE Transactions on, vol. 57, no. 2, pp , Feb [14] I. Wajid, M. Pesavento, Y.C. Eldar, and D. Ciochina, Robust downlink beamforming with partial channel state information for conventional and cognitive radio networks, Signal Processing, IEEE Transactions on, vol. 61, no. 14, pp , July [15] B.K. Chalise, S. ShahbazPanahi, A. Czylwik, and A.B. Gershman, Robust downlink beamforming based on outage probability specifications, Wireless Communications, IEEE Transactions on, vol. 6, no. 10, pp , October [16] C. Masouros and E. Alsusa, Soft linear precoding for the downlink of ds/cdma communication systems, Vehicular Technology, IEEE Transactions on, vol. 59, no. 1, pp , Jan [17] C. Masouros and E. Alsusa, Dynamic linear precoding for the exploitation of known interference in MIMO broadcast systems, Wireless Communications, IEEE Transactions on, vol. 8, no. 3, pp , March [18] C. Masouros, Correlation rotation linear precoding for MIMO broadcast communications, Signal Processing, IEEE Transactions on, vol. 59, no. 1, pp , Jan [19] C. Masouros and T. Ratnarajah, Interference as a source of green signal power in cognitive relay assisted co-existing MIMO wireless transmissions, Communications, IEEE Transactions on, vol. 60, no. 2, pp , February [20] C. Masouros, M. Sellathurai, and T. Ratnarajah, Interference optimization for transmit power reduction in Tomlinson- Harashima precoded MIMO downlinks, Signal Processing, IEEE Transactions on, vol. 60, no. 5, pp , May [21] C. Masouros, T. Ratnarajah, M. Sellathurai, C.B. Papadias, and A.K. Shukla, Known interference in the cellular downlink: a performance limiting factor or a source of green signal power?, Communications Magazine, IEEE, vol. 51, no. 10, pp , October [22] G. Zheng, I. Krikidis, C. Masouros, S. Timotheou, D.-A. Toumpakaris, and Zhiguo Ding, thinking the role of interference in wireless networks, Communications Magazine, IEEE, vol. 52, no. 11, pp , Nov [23] C. Masouros and G. Zheng, Exploiting known interference as green signal power for downlink beamforming optimization, Signal Processing, IEEE Transactions on, vol. 63, no. 14, pp , Jul [24] M. Alodeh, S. Chatzinotas, and B. Ottersten, Constructive multiuser interference in symbol level precoding for the MISO downlink channel, Signal Processing, IEEE Transactions on, vol. 63, no. 9, pp , May [25] G.S. Smith, A direct derivation of a single-antenna reciprocity relation for the time domain, Antennas and Propagation, IEEE Transactions on, vol. 52, no. 6, pp , [26] M. Grant, S. Boyd, and Y. Ye, CVX: Matlab software for disciplined convex programming, Online accessiable: edu/ boyd/cvx, [27] Y. Huang and D.P. Palomar, Rank-constrained separable semidefinite programming with applications to optimal beamforming, IEEE Transactions on Signal Processing, vol. 58, no. 2, pp , Feb